(Exercises for Section 5.1: Antiderivatives and Indefinite Integrals) E.5.1

CHAPTER 5: INTEGRALS

SECTION 5.1: ANTIDERIVATIVES / INDEFINITE INTEGRALS

1) Evaluate the following indefinite integrals. Assume that all integrands are defined [and continuous] “where we care.”

a)

2x3 −x34

5−

x5

2+

23 x

− 3⎛

⎝⎜

⎞

⎠⎟ dx∫

b) y y dy∫

c)

3t3 − 2t2 + tt

dt∫

d) w + 3( ) w + 4( ) dw∫

e) 3z( )4dz∫

f)

t2 + 3( )2

t6dt∫ . Your final answer must not contain negative exponents.

g)

x3 + 8x + 2

dx∫

h)

3sin x + 5cos x( ) dx∫

i)

4cos2 θ

dθ∫

j) sec t( ) csc t( ) cot t( ) dt∫

k)

sin rcos2 r

dr∫

l) cotθ( ) 1+ cot2 θ( ) sinθ( ) dθ∫

m) π 2 dx∫

n) sin5π dx∫

o) a10 + abt( )dt∫ , if a and b represent real constants

p)

Dx tan5 x4( )⎡⎣

⎤⎦ dx∫

(Exercises for Section 5.1: Antiderivatives and Indefinite Integrals) E.5.2

2) Evaluate Dx x5 + x dx∫( ) .

3) For each part below, solve the differential equation subject to the given conditions.

a) ′f x( ) = 6x2 + 2x −1 subject to: f −2( ) = 30

b) dydx

= 2 x subject to: y = 34 if x = 9

c) ′′f x( ) = 3x + 2 subject to: ′f 1( ) = 1

2 and f −1( ) = 7

d)

d 2 ydx2

= 7sin x + 2cos x subject to: both ( y = 10 and ′y = 4 ) if x = 0

4) Assuming that a particle is moving on a coordinate line with acceleration

a t( ) = 2 − 6t and with initial conditions v 0( ) = −5 and s 0( ) = 4 , find the position

function rule s t( ) for the particle. s t( ) is measured in feet, t in minutes, v t( ) in

feet per minute, and a t( ) in feet per minute per minute.

5) On Earth, a projectile is fired vertically upward from ground level with a velocity of 1600 feet per second. Ignore air resistance. Use −32 feet per second per second as the Earth’s [signed] gravitational constant of acceleration.

a) Find the projectile’s height s t( ) above ground t seconds after it is fired. (The formula will be relevant until the time the projectile hits the ground.)

b) Find the projectile’s maximum height.

6) A brick is thrown directly downward from a height of 96 feet with an initial speed of 16 feet per second. Use −32 feet per second per second as the Earth’s [signed] gravitational constant of acceleration.

a) Find the brick’s height above ground t seconds after it is thrown. (Your formula will be relevant up until the time at which the brick hits the ground.)

b) Find how long it takes for the brick to hit the ground (starting from the time the brick is thrown).

c) Find the velocity at which the brick hits the ground.

7) A country has natural gas reserves of 250 billion cubic feet. If the gas is consumed at the rate of 3+ 0.02t billion cubic feet per year (where t = 0 corresponds to now, and t is measured in years), how long will it be before the reserves are depleted? Round off your answer to four significant digits.

(Exercises for Section 5.2: u Substitutions) E.5.3

SECTION 5.2: u SUBSTITUTIONS

1) Evaluate the following indefinite integrals. In this section, assume that all integrands are defined [and continuous] “where we care.”

a) 3x x2 + 5( )3dx∫

b) 4 p − 5 dp∫

c) α 4 sin π − 2α 5( ) dα∫

d)

2r3 + r

r 4 + r 2dr∫

e) sin x( ) sec5 x( ) dx∫

f)

sin 3θ( ) +θ⎡⎣ ⎤⎦ dθ∫

g)

sec2 x( )x( ) tan3 x( ) dx∫

h)

1x2 + 2x +1

dx∫

i)

secβ tanβ4 +12secβ + 9sec2 β

dβ∫

j)

x

x +1dx∫

k) sin2 x dx∫

2) Use “Guess-and-Check” to evaluate the following indefinite integrals.

a)

cos 4x( ) dx∫ ; b) sin 3θ( ) dθ∫ ; c)

sec2 θ2

⎛⎝⎜

⎞⎠⎟

dθ∫ ; d) sec 2α( ) tan 2α( ) dα∫ ;

e) csc 7α( )cot 7α( ) dα∫

3) We will evaluate

x −1( )2

xdx∫ in two different ways.

a) Evaluate

x −1( )2

xdx∫ without using a substitution.

b) Evaluate

x −1( )2

xdx∫ by using a substitution.

c) Show that your answers to a) and b) are equivalent. (This is mostly algebra.)

ADDITIONAL PROBLEMS (#4-6)

4) Evaluate

2 + 1z

⎛⎝⎜

⎞⎠⎟

4

⋅1

z zdz∫ .

5) Evaluate u2 +1( ) u − 2 du∫ . Note: Do not use u as your substitution variable.

6) Evaluate

x2 + 2xx2 + 2x +1

dx∫ .

(Exercises for Sections 5.3/5.4: Area and Definite Integrals) E.5.4

SECTIONS 5.3/5.4: AREA and DEFINITE INTEGRALS

1) Assume that

x dx1

4

∫ =143

. Evaluate the following integrals based on this

information.

a)

t dt1

4

∫

b)

x dx4

4

∫ + x dx4

1

∫

2) Evaluate the following definite integrals using areas.

a)

6 dx−1

5

∫

b)

2x + 6( ) dx−3

2

∫

c)

x −1 dx0

3

∫ . We will revisit this problem in Section 5.6.

d)

9 − x2 dx0

3

∫

e)

3+ 4 − x2( ) dx− 2

2

∫

3) We will approximate

2x + 3( ) dx1

5

∫ using Riemann approximations and the

partition P of the interval 1, 5⎡⎣ ⎤⎦ , where P is given by: 1, 3, 4, 5{ }.

a) Use a Left-Hand Riemann Approximation.

b) Use a Right-Hand Riemann Approximation.

c) Use a Midpoint Riemann Approximation.

d) Use an area argument to find the exact value of the integral.

4) Approximate

8 − 12

x2⎛⎝⎜

⎞⎠⎟

dx0

6

∫ by using a Midpoint Riemann Approximation and a

regular partition of the interval 0, 6⎡⎣ ⎤⎦ into six subintervals.

(Exercises for Section 5.5: Properties of Definite Integrals) E.5.5

SECTION 5.5: PROPERTIES OF DEFINITE INTEGRALS

1) Assuming that

x2 dx1

4

∫ = 21 and

x dx1

4

∫ =152

, evaluate the following definite

integrals.

a)

3x2 + 5( ) dx1

4

∫

b)

2 − 9t − 4t2( ) dt4

1

∫

2) True or False:

1+ sin x( ) dx0

4π

∫ ≥ 0 . Explain.

3) True or False:

−cos2 x( ) dx3π

5π

∫ ≤ 0 . Explain.

4) Express as one integral:

f x( ) dx5

1

∫ + f x( ) dx−3

5

∫ .

Assume that f is everywhere continuous on .

5) Express as one integral:

g t( ) dtc

c+h

∫ − g t( ) dtc

h

∫ .

Assume that g is everywhere continuous on . c and h represent real constants.

6) Let f x( ) = x2 +1. Assume that

f x( ) dx− 2

1

∫ = 6 ; you will be able to verify this

after you study Section 5.6.

a) Find the average value of f on −2,1⎡⎣ ⎤⎦ .

b) Find a number z that satisfies the conclusion of the Mean Value Theorem (MVT) for Integrals, where the interval of interest is −2,1⎡⎣ ⎤⎦ .

7) Let g x( ) = 3 x +1 . Assume that

g x( ) dx−1

8

∫ = 54 ; you will be able to verify this

after you study Section 5.6.

a) Find the average value of g on −1, 8⎡⎣ ⎤⎦ .

b) Find a number z that satisfies the conclusion of the Mean Value Theorem (MVT) for Integrals, where the interval of interest is −1, 8⎡⎣ ⎤⎦ . (Consider g instead of f.)

(Exercises for Section 5.6: Fundamental Theorem of Calculus (FTC)) E.5.6

SECTION 5.6: FUNDAMENTAL THEOREM OF CALCULUS (FTC)

1) Evaluate the following definite integrals.

a)

x2 +1( ) dx− 2

1

∫ . We saw this in Section 5.5, Exercise 6. Yes, the answer is 6.

b)

dt4

9

∫

c)

r + 2r

dr9

16

∫

d)

x23 − 2( ) dx−8

8

∫ . Hint: Do you see how you can reduce your work?

e)

t2 −1t −1

dt4

2

∫

f)

sin4 θ 2( ) dθ5

5

∫ .

Hint: If you’re spending too much time on this, you’re doing it wrong!

g)

5sin x dx0

π

∫

h)

sec t( ) sec t + tan t( ) dt0

π /4

∫

i)

f x( ) dx0

2

∫ , where f x( ) = x, if 0 ≤ x <1

x2 , if 1≤ x ≤ 2⎧⎨⎩

2) Evaluate the following definite integrals by using u substitutions (or maybe “Guess-and-Check” if the u sub is linear).

a)

x2 x3 +1( )3dx

− 2

3

∫

b)

5− x dx1

4

∫

c)

1

3− 2x( )2dx

0

1

∫

d)

1

x x +1( )3dx

1

4

∫

e)

cosθ3

⎛⎝⎜

⎞⎠⎟

dθπ /2

π

∫ .

In e), give an exact answer and also give an approximate answer rounded off to three significant digits.

(Exercises for Section 5.6: Fundamental Theorem of Calculus (FTC)) E.5.7

f)

4sin 2x( ) + 6cos 3x( )⎡⎣ ⎤⎦ dxπ /4

π /3

∫

In f), give an exact answer and also give an approximate answer rounded off to two significant digits.

g)

sin xcos2 x

dx0

π /3

∫

3) We will evaluate

8 − 5x( )2dx

− 2

−1

∫ in two different ways.

a) First expand 8 − 5x( )2, and then integrate as usual.

b) Use a u substitution.

4) In Sections 5.3/5.4, Exercise 2c, you evaluated

x −1 dx0

3

∫ using areas. Now, use

the definition of absolute value and evaluate this integral using the FTC.

5) Evaluate

x + sin x( ) dx−π

π

∫ by finding an antiderivative and applying the FTC as

usual. Why does your answer make sense?

6) Let f x( ) = x

x2 + 9.

a) Find the average value of f on 0, 4⎡⎣ ⎤⎦ .

b) Find a number z that satisfies the conclusion of the Mean Value Theorem (MVT) for Integrals, where the interval of interest is 0, 4⎡⎣ ⎤⎦ .

7) Find Dx sin10 x dx

π /4

π

∫⎛⎝⎞⎠ without performing the integration.

8) Find Dx sin10 t dt

π /4

x

∫⎛⎝⎞⎠ without performing the integration.

(Exercises for Section 5.7: Numerical Approximation of Definite Integrals) E.5.8.

SECTION 5.7: NUMERICAL APPROXIMATION OF DEFINITE INTEGRALS

1) We will find approximations for

11+ x2 dx

1

3

∫ by using a regular partition of the

interval 1, 3⎡⎣ ⎤⎦ into n = 4 subintervals. Round off intermediate results to five decimal places, and round off final answers to four decimal places.

a) Use the Trapezoidal Rule.

b) Use Simpson’s Rule.

Note: The exact value of the integral is tan−1 3( )− π

4≈ 0.4636 . You will learn how

to work out this integral exactly in Chapter 8.

2) We will find approximations for

sin x dx0

π

∫ by using a regular partition of the

interval 0,π⎡⎣ ⎤⎦ into n = 6 subintervals. Round off intermediate results to five decimal places, and round off final answers to four decimal places.

a) Use the Trapezoidal Rule.

b) Use Simpson’s Rule.

Note for Kuniyuki’s class: If related questions are placed on exams, then you will be given a formula for the Trapezoidal Rule and/or Simpson’s Rule, as appropriate.